Positive solutions of nonlinear boundary value problems for delayed fractional q-difference systems

Qiaozhen Yuan,Wengui Yang

doi:10.1186/1687-1847-2014-51

Abstract

In this paper, we investigate the existence and uniqueness of positive solutions to nonlinear boundary value problems for delayed fractional q-difference systems by applying the properties of the Green function and some well-known fixed-point theorems. As applications, some examples are presented to illustrate the main results. MSC:39A13, 34B18, 34A08.

Highlights

In the past decades, fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in various fields of science and engineering such as physics, mechanics, chemistry, biology, engineering, etc
Ahmad and Nieto [ ] dealt with some existence results for a boundary value problem involving a nonlinear fractional order integrodifferential equation with integral boundary conditions based on a contraction mapping principle and Krasnoselskiii’s fixed-point theorem
Function Gi(t, s) defined above satisfies the following conditions: (a) Gi(t, qs) ≥ and Gi(t, qs) ≤ Gi( , qs) for all ≤ t, s ≤ ; (b) Gi(t, qs) ≥ gi(t)Gi( , qs) for all ≤ t, s ≤ with g(t) = tαi–

Summary

Introduction

Fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in various fields of science and engineering such as physics, mechanics, chemistry, biology, engineering, etc. Ahmad and Nieto [ ] dealt with some existence results for a boundary value problem involving a nonlinear fractional order integrodifferential equation with integral boundary conditions based on a contraction mapping principle and Krasnoselskiii’s fixed-point theorem. Ahmad et al [ ] investigated the existence and uniqueness of solutions for a class of Caputo-type fractional boundary value problems involving four-point nonlocal Riemann-Liouville integral boundary conditions of different order by means of standard tools of fixed-point theory and Leray-Schauder nonlinear alternative.

Objectives

Results

Conclusion